Numbers are man's work Gerhard Post, DWMP Mathematisch Café, 17 juni 2013.

Numbers are man's work

Gerhard Post, DWMP

Mathematisch Café, 17 juni 2013

The dear God has made the whole numbers,

all the rest is man's work.

Leopold Kronecker (1823 - 1891)

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Numbers are man's work

Leopold Kronecker Two interwoven stories:• The concept “number”• The representation of a number.

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Egyptian fractions

A Number is a sum of distinct unit fractions,

such as = + +

®

Rhind papyrus (1650 BC)

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Egyptian fractions: construction

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Egyptian fractions: why ?

A possible reason is easier (physical) division:

= +

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The Greek

A Number is a ratio of integers

is not a number

or: a number is a solution to an equation of the form:

c1 x + c0 = 0 (c1 and c0 integers)

Hippasus (5th century BC) is believed

to have discovered that is not a

number

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The Greek (after Hippasus)

A Number is a solution to an equation of the form:

cn x n + cn-1 x

n-1 + … + c1 x + c0 = 0

for integers cn ,…,c0.

®

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Orloj, Prague (15th century)

Orloj - Astronomical Clock - Prague

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Orloj, Prague (15th century)

Toothed wheels

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Orloj, Prague

A Number is a ratio of ‘small’ integers®

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Orloj, Prague

A Number is a ratio of ‘small’ integers®

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Orloj, Prague

How to construct these small integers ?

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The Italians (Cardano’s “Ars Magna”, 1545)

A Number is a solution to an equation of the form:

cn x n + cn-1 x

n-1 + … + c1 x + c0 = 0

®

Girolamo Cardano Niccolò Tartaglia Lodovico Ferrari

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Solve: x 3 + a x

2 + b x + c = 0

1. Replace x by (x a) (drop the prime) gets rid of x 2 :

2. Substitute u - v for x

3. Take 3u v = b:

4. Substitute v = 1/3 b/u → quadratic equation in u3.

x 3 + b x + c = 0

(u 3 3uv (u v) v

3) + b (u v) + c = 0

u 3 v

3 + c = 0

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Simon Stevin Brugensis (1548 1620)

A Number is a decimal expansion

Simon Stevin

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Beginning of 19th century

A Number is an algebraic number (since 500 BC)®

An algebraic number is a solution to an equation of the form:

cn x n + cn-1 x

n-1 + … + c1 x + c0 = 0

for integers cn ,…,c0.

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Joseph Liouville (1809 - 1882)

f(x) = cn x n + cn-1 x n-1 + … + c1 x + c0 = 0

(integers cn ,…, c0).

If is an irrational algebraic number satisfying f ()=0 the equation above, then there exists a number A > 0 such that, for all integers p and q with q > 0:

The key observation to prove this is: |f ()| if f () ≠ 0,and ) )

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Joseph Liouville (1809 - 1882)

A Liouville number is a number with the property that, for every positive integer n, there exist integers p and q with q > 0 and such that

0 <

A Number is an algebraic or a Liouville number®

Joseph Liouville

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Joseph Liouville (1809 - 1882)

Liouville’s constant: + … = 0.11000100000000000000000100…

Q: How many Liouville numbers are there?

A: As many as all decimal expansions…

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Georg Cantor (1845 –1918)

A Number is a decimal expansion®

Not all infinities are the same

Leopold Kronecker: “I don't know what predominates in Cantor's theory – philosophy or theology, but I am sure that there is no mathematics there.”

David Hilbert: “No one will drive us from the paradise which Cantor created for us.”

Georg Cantor

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Conclusions

A Number is …®

Although the numbers are man’s work,they brought us to paradise…